Local gaps in three-dimensional periodic media

We consider the propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape. The inclusion size $a$ is much smaller than the array period. We show that global gaps do not exist if $a$ is small enough. The notion of local gaps which depends on the choice of the wave vector $\text{bk}$ is introduced and studied. We determine analytically the location of local gaps for the Dirichlet and transmission problems.